A recent result on the potential of ΔΣ modulators (ΔΣMs) as heuristic optimizers for circulant unconstrained discrete quadratic programming (C-UDQP) problems is revisited, bridging it with current developments on the design of ΔΣMs by semi-definite programming (SDP). This provides an efficient strategy by which one can design a ΔΣM and its input signal from a C-UDQP specification so that the solution of the C-UDQP problem can be found in the ΔΣMs output, all with little manual intervention. The proposed concept is validated by simulation- based experiments of a benchmark case, comparing the new strategy to previous results and exact optimization techniques.